Close six faces without inventing a local field
Gauss law relates a closed-surface flux to net enclosed charge. It does not say every field on that surface is uniform, nor that zero net charge means zero field. A useful test case is an affine field whose constant offsets affect individual faces but cancel from their closed sum.
Learning outcome: Change a and b separately, predicting which face pairs change. Retain all six signed terms rather than only the sum. Find two different settings with zero net flux and show that their point fields can differ.
Starting recap
Review oriented normals, definite integrals, source volume measures and partial derivatives as one-coordinate rates. Potential differences use an explicit reference and field work uses a signed line integral.
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For a=1 and b=0, the six outward fluxes are −3,5,4,−4,−2,0 N m²/C, summing to zero. Yet at the origin E=(3,−2,1) N/C and magnitude √14 N/C. Zero enclosed net charge therefore does not establish zero field. At a=2,b=1 the closed flux is 4 N m²/C.
Open the experimentPredict
What follows from zero closed electric flux without additional symmetry?
Experiment
Change a and b separately, predicting which face pairs change. Retain all six signed terms rather than only the sum. Find two different settings with zero net flux and show that their point fields can differ.
Close six oriented faces without inferring a uniform field
Worked teaching model · synthetic values · Box 0≤x≤2 m,0≤y,z≤1 m with outward normals. Prescribed electrostatic field E=(a x+3,b y−2,1−z) in consistent SI, hence curl zero. Nonzero divergence implies stipulated local charge density. Uniform offsets affect face fluxes but cancel in closed flux. Rounded K=9×10⁹ fixes epsilon0=1/(4πK) consistently.
Reset keeps your prediction, captures and practice history.
East field through the two opposing faces
Unlike graph quantities rescale independently; read units and values, not screen angles. 1: prescribed affine field (solid). Joined curves show the supplied model, not additional observations.
Equivalent numerical coordinates for East field through the two opposing faces
prescribed affine field
| x coordinate (m) | east field (N/C) |
|---|---|
| 0 | 3 |
| 0.025 | 3.025 |
| 0.05 | 3.05 |
| 0.075 | 3.075 |
| 0.1 | 3.1 |
| 0.125 | 3.125 |
| 0.15 | 3.15 |
| 0.175 | 3.175 |
| 0.2 | 3.2 |
| 0.225 | 3.225 |
| 0.25 | 3.25 |
| 0.275 | 3.275 |
| 0.3 | 3.3 |
| 0.325 | 3.325 |
| 0.35 | 3.35 |
| 0.375 | 3.375 |
| 0.4 | 3.4 |
| 0.425 | 3.425 |
| 0.45 | 3.45 |
| 0.475 | 3.475 |
| 0.5 | 3.5 |
| 0.525 | 3.525 |
| 0.55 | 3.55 |
| 0.575 | 3.575 |
| 0.6 | 3.6 |
| 0.625 | 3.625 |
| 0.65 | 3.65 |
| 0.675 | 3.675 |
| 0.7 | 3.7 |
| 0.725 | 3.725 |
| 0.75 | 3.75 |
| 0.775 | 3.775 |
| 0.8 | 3.8 |
| 0.825 | 3.825 |
| 0.85 | 3.85 |
| 0.875 | 3.875 |
| 0.9 | 3.9 |
| 0.925 | 3.925 |
| 0.95 | 3.95 |
| 0.975 | 3.975 |
| 1 | 4 |
| 1.025 | 4.025 |
| 1.05 | 4.05 |
| 1.075 | 4.075 |
| 1.1 | 4.1 |
| 1.125 | 4.125 |
| 1.15 | 4.15 |
| 1.175 | 4.175 |
| 1.2 | 4.2 |
| 1.225 | 4.225 |
| 1.25 | 4.25 |
| 1.275 | 4.275 |
| 1.3 | 4.3 |
| 1.325 | 4.325 |
| 1.35 | 4.35 |
| 1.375 | 4.375 |
| 1.4 | 4.4 |
| 1.425 | 4.425 |
| 1.45 | 4.45 |
| 1.475 | 4.475 |
| 1.5 | 4.5 |
| 1.525 | 4.525 |
| 1.55 | 4.55 |
| 1.575 | 4.575 |
| 1.6 | 4.6 |
| 1.625 | 4.625 |
| 1.65 | 4.65 |
| 1.675 | 4.675 |
| 1.7 | 4.7 |
| 1.725 | 4.725 |
| 1.75 | 4.75 |
| 1.775 | 4.775 |
| 1.8 | 4.8 |
| 1.825 | 4.825 |
| 1.85 | 4.85 |
| 1.875 | 4.875 |
| 1.9 | 4.9 |
| 1.925 | 4.925 |
| 1.95 | 4.95 |
| 1.975 | 4.975 |
| 2 | 5 |
At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.
| Quantity | Value | Unit |
|---|---|---|
| Left x face outward flux | -3 | N m²/C |
| Right x face outward flux | 5 | N m²/C |
| Lower y face outward flux | 4 | N m²/C |
| Upper y face outward flux | -4 | N m²/C |
| Lower z face outward flux | -2 | N m²/C |
| Upper z face outward flux | 0 | N m²/C |
| Total closed flux | 0 | N m²/C |
| Local divergence | 0 | N/C/m |
| Inferred uniform volume charge density | 0 | C/m³ |
| Enclosed charge | 0 | C |
| Field magnitude at origin | 3.741657 | N/C |
Zero enclosed net charge or zero closed flux does not imply zero local electric field.
Outward normals reverse the signs on the left, lower-y and lower-z faces. Summing gives 2(a+b−1), equal to volume 2 m³ times divergence a+b−1. Uniform field offsets cancel from net flux but remain in local fields and individual face fluxes. Gauss law constrains an integral; without additional symmetry it does not authorize replacing a varying field by flux divided by total area.
Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.
Zero enclosed net charge or zero closed flux does not imply zero local electric field.
Outward normals reverse the signs on the left, lower-y and lower-z faces. Summing gives 2(a+b−1), equal to volume 2 m³ times divergence a+b−1. Uniform field offsets cancel from net flux but remain in local fields and individual face fluxes. Gauss law constrains an integral; without additional symmetry it does not authorize replacing a varying field by flux divided by total area.
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